Continuity of logarithmic capacity
نویسندگان
چکیده
We prove the continuity of logarithmic capacity under Hausdorff convergence uniformly perfect planar sets. The holds when distance to limit set tends zero at sufficiently rapid rate, compared decay parameters involved in condition. may fail otherwise.
منابع مشابه
Logarithmic estimates for continuity equations
The aim of this short note is twofold. First, we give a sketch of the proof of a recent result proved by the authors in the paper [7] concerning existence and uniqueness of renormalized solutions of continuity equations with unbounded damping coe cient. Second, we show how the ideas in [7] can be used to provide an alternative proof of the result in [6, 9, 12], where the usual requirement of bo...
متن کاملNumerical approximation of the logarithmic capacity
The logarithmic capacity of compact sets in R2 plays an important role in various fields of applied mathematics. Its value can be computed analytically for a few simple sets. In this paper a new algorithm is presented that numerically approximates the logarithmic capacity for more involved sets. The algorithm requires the solution of a boundary integral equation with Dirichlet boundary data. Th...
متن کاملEigenvalue Clustering, Control Energy, and Logarithmic Capacity
We prove two bounds showing that if the eigenvalues of a matrix are clustered in a region of the complex plane then the corresponding discrete-time linear system requires significant energy to control. An interesting feature of one of our bounds is that the dependence on the region is via its logarithmic capacity, which is a measure of how well a unit of mass may be spread out over the region t...
متن کاملAn isoperimetric inequality for logarithmic capacity
We prove a sharp lower bound of the form capE ≥ (1/2)diamE · Ψ(areaE/((π/4)diam 2E)) for the logarithmic capacity of a compact connected planar set E in terms of its area and diameter. Our lower bound includes as special cases G. Faber’s inequality capE ≥ diamE/4 and G. Pólya’s inequality capE ≥ (areaE/π)1/2. We give explicit formulations, functions of (1/2)diamE, for the extremal domains which...
متن کاملAn isoperimetric inequality for logarithmic capacity of polygons
We verify an old conjecture of G. Pólya and G. Szegő saying that the regular n-gon minimizes the logarithmic capacity among all n-gons with a fixed area.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Mathematical Analysis and Applications
سال: 2022
ISSN: ['0022-247X', '1096-0813']
DOI: https://doi.org/10.1016/j.jmaa.2021.125585